It's not trivial if you have to prove this "with bare hands"; see Hurkyl's answer.

But it's trivial if you have a pocket calculator at your disposal. So I'm interpreting the phrase as follows: The author was on his way to heavier and more important things, and he didn't want to interrupt his argument with a proof of this little fact. Depending on circumstances and the envisaged audience he could have made a "Lemma" out of it.

+1. "Trivial" does not mean "obvious" or "quickly proven". It means that, whatever amount of work is needed for the proof, it is uninteresting grunt work. Of course, being obvious makes a proof also trivial, but people forget that not everything trivial is also obvious.
–
rumtschoJan 28 '14 at 18:43

Taking reciprocals of both sides gives the equivalent statement
$$
\left(1+\frac{2\ln3}{3}\right)^{3/2}\ge\frac{3}{2},
$$
which is fairly obvious once you realize that $\ln3>1.$ The latter follows from the fact that $e<3,$ which is not hard to see.